Operadores Unitários em Computação Quântica

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5.5 An Epistemic Quantum Computational Semantics 97
• pT
1 (A|0〉T) = pT
1 (|0〉T) = 0; pT
1 (A|1〉T) = pT
1 (|1〉T) = 1.
Hence, b = 0, d = eıη (for some η ∈ R) and c = 0, a = eıθ (for some θ ∈ R).
Thus,
A =
[
eıθ 0
0 eıη
]
.
2. Suppose that
A =
[
eıθ 0
0 eıη
]
.
Consider a generic qubit |ψ〉 = a0|0〉T + a1|1〉T of H (1). We have:
A|ψ〉 = a0e
ıθ |0〉T + a1e
ıη|1〉T.
Thus,
pT
1 (A|ψ〉) = |a1eıη|2 = |a1|2 = pT
1 (|ψ〉).
Hence, A is a T-probabilistic identity.
�
Theorem 5.4 Let U be a unitary operator ofH (1) such that for any qubit |ψ〉 of the
space,
pT
1 (U|ψ〉) ≤ pT
1 (|ψ〉).
Then, U is a T-probabilistic identity ofH (1).
Proof Let
U =
[
a b
c d
]
.
Thus, U|0〉T = a|0〉T + c|1〉T and U|1〉T = b|0〉T + d|1〉T.
By hypothesis we have:
pT
1 (U|0〉T) = pT
1 (a|0〉T + c|1〉T) ≤ pT
1 (|0〉T) = 0.
Hence, pT
1 (U|0〉T) = 0, c = 0 and a = eıθ (for some θ ∈ R). Consequently,
U =
[
eıθ b
0 d
]
.
Since U is unitary, we have: UU† = I(1). Thus,
[
eıθ b
0 d
] [
(eıθ )∗ 0
b∗ d∗
]
=
[
1 0
0 1
]
.